Robustness and U.S. Monetary Policy Experimentation · this ‘Bellman versus Lucas’ difference of opinion seems to require challenging the Bellman equation that leads to the recommendation

Robustness and U.S. Monetary Policy

Experimentation∗

Timothy Cogley†

Riccardo Colacito‡

Lars Peter Hansen§

Thomas J. Sargent¶

August 8, 2008

Abstract

We study how a concern for robustness modifies a policy maker’s incentiveto experiment. A policy maker has a prior over two submodels of inflation-unemployment dynamics. One submodel implies an exploitable trade-off, theother does not. Bayes’ law gives the policy maker an incentive to experiment.The policy maker fears that both submodels and his prior probability distri-bution over them are misspecified. We compute decision rules that are robustto misspecifications of each submodel and of the prior distribution over sub-models. We compare robust rules to ones that Cogley, Colacito, and Sargent(2007) computed assuming that the models and the prior distribution are cor-rectly specified. We explain how the policy maker’s desires to protect againstmisspecifications of the submodels, on the one hand, and misspecifications ofthe prior over them, on the other, have different effects on the decision rule.

Key words: Learning, model uncertainty, Bayes’ law, Phillips curve, experimen-tation, robustness, pessimism, entropy.

∗We thank Klaus Neusser and a referee for very thoughtful comments on an earlier draft.†University of California, Davis. Email: [email protected]‡The University of North Carolina at Chapel Hill, Kenan-Flagler Business School. Email:

[email protected]§University of Chicago. Email: [email protected]¶New York University and Hoover Institution. Email: [email protected]

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1 Introduction

Central bankers frequently emphasize the importance of taking parameter and/ormodel uncertainty into account when making decisions (e.g., see Greenspan 2004 andM. King 2004). A natural way to do so is to cast an optimal policy problem as aBayesian decision problem. One consequence is that the decision maker’s posteriorabout parameters and model probabilities becomes part of the state vector. A Bell-man equation would then instruct the decision maker to experiment with an eyetoward tightening that posterior in the future. Although experimentation wouldcause near-term outcomes to deteriorate, it would speed learning and improve out-comes in the long run. Cogley, Colacito, and Sargent (2007), Wieland (2000a,b),and Beck and Wieland (2002) study aspects of this tradeoff in a variety of mone-tary policy models. Whether the decision maker should experiment a little or a lotdepends on details of the model, but all such studies agree that an optimal policyshould include some experimentation.

Despite this, prominent macroeconomists like Blinder (1998) and Lucas (1981)have forcefully recommended against purposefully experimenting on real economiesin order to refine the policy maker’s knowledge (see Cogley, Colacito, and Sargent(2007) for quotations of Blinder and Lucas). An aversion to experimentation alsoruns through Friedman’s advocacy of a k-percent money growth rule. Resolvingthis ‘Bellman versus Lucas’ difference of opinion seems to require challenging theBellman equation that leads to the recommendation to experiment purposefully.

That is what we do in this paper. In particular, we challenge the ingredient of theBellman equation that specifies that the policy maker completely trusts his stochas-tic specification.1 Our decision maker distrusts his stochastic specification and thismodifies his Bellman equation. We formulate distrust by using risk-sensitivity op-erators and we study how that alters incentives to experiment.

As a laboratory, we adopt the model of Cogley, Colacito, and Sargent (2007).That paper computed the benefits to a Bayesian decision maker of intentional ex-perimentation designed to reduce uncertainty about the correct model specification.The authors gave a policy maker two submodels that have very different operatingcharacteristics that are important for policy. They also assumed that the monetaryauthority’s doubts are limited to not knowing the ‘correct’ value of one hyperparam-eter, α, the probability that one of two competing submodels generates the data.In other words, they assumed that the monetary authority has narrowed the setof possible models down to two and that it knows each submodel perfectly. If inpractice one thinks that the monetary authority’s doubts are broader and vaguer,

1Thus, Marimon (1997) noted that a Bayesian ‘knows the truth’ from the outset, so thatBayesian learning problems are just about conditioning, not constructing new joint distributionsover unknowns and data.

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then their calculations substantially understated the difficulty of the decision prob-lem confronting the policy maker. For instance, the decision maker might be unsureabout parameters of each submodel, might suspect that additional submodels arerelevant, and might also have qualms about whether his prior adequately representshis beliefs.2 The robustness calculations in this paper are designed to address someof these concerns. As we shall see, a robust decision maker still has an incentiveto experiment, but the degree of experimentation is tempered by concerns that thedecision problem is misspecified.

We use two risk-sensitivity operators defined by Hansen and Sargent (2005,2007a) to construct a Bellman equation that acknowledges that the policy makerdistrusts his model specification wants a decision rule that will be good enough de-spite the misspecification of his model. ‘Good enough’ means that a decision ruleattains an acceptable outcome for a set of stochastic specifications centered on thepolicy maker’s baseline model. As we shall see, our risk-sensitivity operators sum-marize how the policy maker does a worst-case analysis in order to design a robustrule.

Our robust policy maker achieves a robust decision rule by pretending to be apessimist. But pessimistic about what? Any worst-case analysis is context specificin the sense that ‘worst’ is relative to a particular objective function. Our decisionmaker attains robustness by finding a worst-case rule for a particular Kydland-Prescott (1977) ad hoc criterion for assessing inflation and unemployment outcomepaths. As we vary the weights on inflation and unemployment in that criterion,what is worst changes. That affects the robust decision rule in ways that we areabout to study.

1.1 Organization

Section 2 formulates a Bellman equation without concerns about misspecifica-tion. Section 3 reformulates the Bellman equation to reflect how the decision makerresponds to fears that his prior over the two submodels as well as the submodelsthemselves are misspecified. Section 4 describes our quantitative findings. Section 5adds some concluding remarks. We consign many technical details to an appendix.

2O’Hagan (1998, p. 22) states that “to elicit a genuine prior distribution (and typically what isneeded is a joint distribution in several dimensions) is a complex business demanding a substantialeffort on the part of both the statistician and the person whose prior beliefs are to be elicited.”Applied Bayesians frequently take shortcuts, such as assuming that parameters are independenta priori or choosing functional forms for convenience and not from conviction. Consequently, onemight question whether a prior probability model accurately reflects the decision maker’s initialbeliefs.

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2 The experimentation problem without model

ambiguity

A decision maker wants to maximize the following function of states st andcontrols vt:

E0

∞∑

t=0

βtr(st, vt). (1)

The observable and unobservable components of the state vector, st and zt, respec-tively, evolve according to a law of motion

st+1 = πs(st, vt, zt, ǫt+1), (2)

zt+1 = zt, (3)

where ǫt+1 is an i.i.d. vector of shocks and zt ∈ 1, 2 is a hidden state variablethat indexes submodels. Since the state variable zt is time invariant, specification(2)-(3) states that one of the two submodels governs the data for all periods. Butzt is unknown to the decision maker. The decision maker has a prior probabilityProb(z = 1) = α0. Where st = st, st−1, . . . , s0, the decision maker recursivelycomputes αt = Prob(z = 1|st) by applying Bayes’ law:

αt+1 = πα(αt, πs(st, vt, zt, ǫt+1)). (4)

Because he does not know zt, the policy maker’s prior probability αt becomes astate variable in a Bellman equation that captures his incentive to experiment. Letasterisks denote next-period values and express the Bellman equation as

V (s, α) = maxv

r(s, v) + Ez

[Es∗,α∗(βV (s∗, α∗)|s, v, α, z)|s, v, α

], (5)

subject to

s∗ = πs(s, v, z, ǫ∗), (6)

α∗ = πα(α, πs(s, v, z, ǫ∗)). (7)

Ez denotes integration with respect to the distribution of the hidden state z thatindexes submodels, and Es∗,α∗ denotes integration with respect to the joint distri-bution of (s∗, α∗) conditional on (s, v, α, z).

3 Experimentation with model ambiguity

Bellman equation (5) invites us to consider two types of misspecification of thestochastic structure: misspecification of the distribution of (s∗, α∗) conditional on

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(s, v, α, z), and misspecification of the probability α over submodels z. FollowingHansen and Sargent (2005, 2007a), we introduce two risk-sensitivity operators thatcan help the decision maker construct a decision rule that is robust to these typesof misspecification. While we refer to them as “risk-sensitivity” operators, it isactually their dual interpretations that interest us. Under these dual interpretations,a risk-sensitivity adjustment is an outcome of a minimization problem that assignsworst-case probabilities subject to a penalty on relative entropy. Thus, we view theoperators as adjusting probabilities in cautious ways that assist the decision makerdesign robust policies.3

3.1 Two risk-sensitivity operators

3.1.1 T1 operator

The risk-sensitivity operator T1 helps the decision maker guard against misspec-

ification of a submodel. Let W (s∗, α∗) be a measurable function of (s∗, α∗). In ourapplication, W will be a continuation value function. Instead of taking conditionalexpectations of W , we shall apply the operator:

T1(W (s∗, α∗))(s, α, v, z; θ1) = −θ1 logEs∗,α∗ exp

(−W (s∗, α∗)

θ1

)∣∣∣(s, α, v, z). (8)

This operator yields the indirect utility function for a problem in which the decisionmaker chooses a worst-case distortion to the conditional distribution for (s∗, α∗) inorder to minimize the expected value of a value function W plus an entropy penalty.That penalty limits the set of alternative models against which the decision makerguards. The size of that set is constrained by the parameter θ1 and is decreasing inθ1, with θ1 = +∞ signifying the absence of a concern for robustness. The solutionto this minimization problem implies a multiplicative distortion to the Bayesianconditional distribution over (s∗, α∗). The worst-case distortion is proportional to

exp(−W (s∗, α∗)

θ1

), (9)

where the factor of proportionality is chosen to make this non-negative randomvariable have conditional expectation equal to unity. Notice that the scaling factorand the outcome of applying the T

1 operator will depend on the state z indexingsubmodels even though W does not. In appendix A, we discuss in more detail aformula for this worst-case conditional distribution. Notice how (9) pessimisticallytwists the conditional density of (s∗, α∗) by upweighting outcomes that have lowervalue.

3Direct motivations for risk sensitivity can be found in Kreps and Porteus (1978) and Klibanoff,Marinacci, and Mukerji (2005).

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3.1.2 T2 operator

The risk-sensitivity operator T2 helps the decision maker evaluate a continuation

value function U that is a measurable function of (s, α, v, z) in a way that guardsagainst misspecification of his prior α:

T2(W (s, α, v, z))(s, α, v; θ2) = −θ2 logEz exp

(−W (s, α, v, z)

θ2

)∣∣∣(s, α, v) (10)

This operator yields the indirect utility function for a problem in which the decisionmaker chooses a distortion to his Bayesian prior α in order to minimize the expectedvalue of a function W (s, α, v, z) plus an entropy penalty. Once again, that penaltyconstrains the set of alternative specifications against which the decision makerwants to guard, with the size of the set decreasing in the parameter θ2. The worst-case distortion to the prior over z is proportional to

exp(−W (s, α, v, z)

θ2

), (11)

where the factor of proportionality is chosen to make this nonnegative random vari-able have mean one. The worst-case density distorts the Bayesian probability byputting higher probability on outcomes with lower continuation values. See appendixA for more details about the worst-case density for z.4

Our decision maker directly distorts the date t posterior distribution over thehidden state, which in our example indexes the unknown model, subject to a penaltyon relative entropy. The source of this distortion could be a change in a priordistribution at some initial date or it could be a past distortion in the state dynamicsconditioned on the hidden state or model.5 Rather than being specific about thissource of misspecification and updating all of the potential probability distributionsin accordance with Bayes rule with the altered priors or likelihoods, our decisionmaker directly explores the impact of changes in the posterior distribution on hisobjective.

4The worst-case model as we have depicted it will depend on the endogenous state variablest. Since this worst-case model distorts the distribution of ǫt+1, we may prefer to represent thisdistortion without explicit dependence on an endogenous state variable. This can often be donefor decision problems without hidden states using a ‘Big K, little k’ argument of a type featuredin chapters 7 and 12 of Hansen and Sargent (2007b). A more limited notion of a worst-case modelcan be constructed when hidden states are present, as discussed in Hansen and Sargent (2007a).

5A change in the state dynamics would imply a misspecification in the evolution of the stateprobabilities.

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3.2 A Bellman equation for inducing robust decision rules

Following Hansen and Sargent (2005, 2007a), we induce robust decision rulesby replacing the mathematical expectations in (5) with risk-sensitivity operators.In particular, we substitute (T1)(θ1) for Es∗,α∗ and replace Ez with (T2)(θ2). Thisdelivers a Bellman equation

V (s, α) = maxv

r(s, v) + T

2[T

1(βV (s∗, α∗)(s, v, α, z; θ1))](s, v, α; θ2)

. (12)

We find it convenient to separate the two risk-sensitivity operators by allow-ing for the parameters θ1 an θ2 to differ. The T

1 operator explores the impactof forward-looking distortions in the state dynamics and the T

2 operator exploresbackward-looking distortions in the outcome of predicting the current hidden stategiven current and past information. As we will see, applications of these two oper-ators have very different ramifications for experimentation, and for that reason wefind it natural to explore them separately.6

3.3 The submodels

Each submodel of Colacito, Cogley, and Sargent (2007) has the form

st+1 = Azst +Bzvt + Czǫt+1, (13)

z = 1, 2, where st is an observable state vector, vt is a control vector, and ǫt+1 is ani.i.d. Gaussian processes with mean zero and contemporaneous covariance matrix I.Let F (·) denote the c.d.f. of this normalized multivariate Gaussian distribution. Attime t, the policy maker has observed a history of outcomes st = st, st−1, . . . , s0 andassigns probability αt to model 1 and probability (1 − αt) to model 2.

To capture an old debate between advocates of the natural unemployment hy-pothesis and those who thought that there was an exploitable unemployment-inflationtrade-off, we imagine that a monetary policy authority has the following two modelsof inflation-unemployment dynamics:7

• Model 1 (Samuelson-Solow):

Ut = .0023 + .7971Ut−1 − .2761πt + .0054η1,t

πt = vt−1 + .0055η3t

6When θ1 = θ2 the two operators applied in conjunction give the recursive formulation of risksensitivity proposed in Hansen and Sargent (1995), appropriately modified for the inclusion ofhidden states.

7We use these specifications in order to have good fitting models, to keep the dimension of thestate to a minimum, and still to allow ourselves to represent ‘natural rate’ and ‘non-natural rate’theories of unemployment. For details, see appendix D of Cogley, Colacito, and Sargent.

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• Model 2 (Lucas):

Ut = .0007 + .8468Ut−1 − .2489(πt − vt−1) + .0055η2,t

πt = vt−1 + .0055η4t

Ut is the deviation of the unemployment rate from an exogenous measure of a naturalrate U∗

t , πt is the quarterly rate of inflation, vt−1 is the rate of inflation that at time t−1 the monetary authority and private agents had both expected to prevail at time t,and, for i = 1, 2, 3, 4, ηit are i.i.d. Gaussian sequences with mean zero and variance 1.The monetary authority has a Kydland-Prescott (1977) loss function E0

∑∞

t=0 βtrt,

where rt = −.5(U2t + λv2

t ) and E0 is the mathematical expectation conditionedon s0, α0. The monetary authority sets vt as a function of time t information.The analysis of Cogley, Colacito, and Sargent (2007) assumed that the monetaryauthority knows the parameters of each model for sure and attaches probabilityα0 to model 1 and probability 1 − α0 to model 2.8 Although they fit the U.S.data from 1948:3-1963:I almost equally well, these two models call for very differentpolicies toward inflation under our loss function. Model 1, whose main featuresmany have attributed to Samuelson and Solow (1960), has an exploitable tradeoffbetween vt and subsequent levels of unemployment. Having operating characteristicsadvocated by Lucas (1972, 1973) and Sargent (1973), model 2 has no exploitablePhillips curve: variations in the predictable part of inflation vt affect inflation butnot unemployment. If α0 = 0, our decision maker should implement the trivialpolicy vt = 0 for all t.9 However, if α0 > 0, the policy maker is willing to set vt 6= 0partly to exploit a probable inflation-unemployment tradeoff and partly to refine α.

Cogley, Colacito, and Sargent (2007) study how decision rules for this problemvary with different values of the decision maker’s preference parameter λ. By com-paring the decision rules from (14) with those from an associated ‘anticipated utility’model, they provide a way to quantify the returns from experimentation.

4 Quantitative findings

4.1 Decision rules without robustness

As a benchmark, we first display the value function and decision rules for aversion of the model without robustness (i.e., for θ1 = θ2 = +∞). Figures 1 and2 depict results for λ = 0.1 and β = 0.995, the parameters favored by Cogley,Colacito, and Sargent (2007). Notice that the value function slopes upward along

8As we shall see below, the T1 operator that we use in section 4.5 allows us to analyze robustness

to model perturbations that can be interpreted as coefficient uncertainty.9Cogley, Colacito, and Sargent adopt a timing protocol that eliminates the inflationary bias.

8

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−0.02

−0.01

0

0.01

0.02

0.03−0.035

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

Prior on Samuelson and Solow (α)Unemployement (U)

Val

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Figure 1: Value function V (U, α) without robustness for λ = 0.1.

−0.01 0 0.01 0.02 0.03−0.02

0

0.02

0.04

0.06α ≈ 0

−0.01 0 0.01 0.02 0.03−0.02

0

0.02

0.04

0.06α = 0.2

−0.01 0 0.01 0.02 0.03−0.02

0

0.02

0.04

0.06α = 0.40

−0.01 0 0.01 0.02 0.03−0.02

0

0.02

0.04

0.06α = 0.60

−0.01 0 0.01 0.02 0.03−0.02

0

0.02

0.04

0.06α = 0.80

−0.01 0 0.01 0.02 0.03−0.02

0

0.02

0.04

0.06α ≈ 1

Figure 2: Decision rules without robustness. Black lines represent optimal experi-ments, and the linear red lines indicate the anticipated-utility approximations de-fined in Cogley, Colacito, and Sargent (2007).

9

the α-axis. Since α is the probability that the Samuelson-Solow model is true, theupward-sloping value function means that the policy maker is better off inhabitinga Keynesian than a classical world. That is because the Samuelson-Solow modelprovides a lever for controlling unemployment that the Lucas model does not. Theinability to control unemployment is costly when λ = 0.1 because in that case thepolicy maker cares a lot about unemployment.

Also notice that for most α the decision rule for programmed inflation slopes up-ward along the U -axis, reflecting the countercyclical nature of policy. In addition,the policy rules are approximately linear, which signifies that there is only a modestincentive to experiment. If the connection between current actions and future beliefswere disregarded, as they are in the anticipated-utility models of Cogley, Colacito,and Sargent (2007), there would be no incentive to experiment, and the problemwould reduce to a linear-quadratic dynamic program, implying linear decision rules.(The linear decision rules displayed in figure 2 are the anticipated utility decisionrules.) The presence of α in the state vector breaks certainty equivalence and makesdecision rules nonlinear, but in our example there is only a slight departure fromlinearity. Optimal monetary-policy experiments involve small, opportunistic per-turbations to programmed inflation relative to anticipated-utility policies, not greatleaps.

4.2 Activating T2 only: robustness with respect to the prior

Next we activate a concern for robustness by reducing θ2 to 0.1. We chose thisvalue partly because it has a noticeable influence on decision rules. The left panelof figure 3 plots the worst-case distortion to αt derived formally in appendix A, andthe right panel plots a pair of decision rules for inflation vt as a function of (Ut, αt).Robust decision rules are shown in red and Bayesian decision rules in gray.

A robust policy maker updates α with Bayes’s theorem, then twists by increasingthe probability weight on the worst-case submodel. The left panel compares theworst-case probability α with the Bayesian probability α. On the boundaries whereα is 0 or 1, α = α. Concerns that the prior is misspecified are irrelevant when thereis no model uncertainty. When α lies between 0 and 1, the worst-case model weightα is always smaller than the Bayesian update α. Since α is the probability attachedto the Samuelson-Solow model, the policy maker twists by reducing his prior weighton that submodel and increasing the probability on the Lucas model. This reflectsthat the policy maker is worse off if the Lucas model is true because then he lacks aninstrument for damping fluctuations in unemployment. Thus, it is understandablethat a policy maker who cares a lot about unemployment will seek robustness bysetting α less than α.

10

0

0.5

1

−0.01

0

0.01

0.02

0.03

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

αt

Ut

d(U t,α

t)

α ≈ 0

0

0.02

0.04

α = 0.2

Infla

tion

α = 0.4

0.00

0.02

0.04

α = 0.6

Infla

tion

−.015 0 .015 .03

α = 0.8

Unemployment−.015 0 .015 .03

0.00

0.02

0.04

α ≈ 1

Unemployment

Infla

tion

Figure 3: Robust policy with T2 operator only, with λ = 0.1 and θ2 = .1. In the

right panel, the grey line is the decision rule for the θ2 = +∞ no-robustness decisionrule, while the dark line is the θ2 = 0.1 robust decision rule.

The right-hand panel of figure 3 shows how concerns about robustness withrespect to the prior over submodels alter the policy rule. Robustness matters mostfor intermediate values of α and high values of |U |. When α is close to 0 or 1, there islittle model uncertainty and therefore little reason to worry about having the wrongmodel weight. In that case, the robust policy closely tracks the original decisionrule. Similarly, the robust rule closely tracks the Bayesian policy when U is close tothe point where the Samuelson-Solow model recommends zero inflation.10 In thatneighborhood of U , the two models recommend similar actions, and since there islittle disagreement, there is also little reason to worry about α. Robustness mattersmore when the models recommend very different actions, i.e. when |U | is large. Forintermediate values of α but high values of U , the robust decision maker sets a lowerinflation target than does one who has no doubts about his prior probabilities. Thisis because the policy maker makes robust decisions by in effect increasing the priorweight that he attaches to the Lucas model, under which inflation is ineffective as atool for affecting unemployment. The analysis is analogous for negative values of U ,

10This occurs when U is slightly less than zero. When U = 0, the Samuelson-Solow modelrecommends a small, positive inflation rate.

11

for the robust policy maker continues to twist by edging the inflation target towardzero.

Comparing these outcomes with section 5.2 of Cogley, Colacito, and Sargent(2007) shows that by expressing distrust of his prior distribution over submodels,application of the T

2 operator diminishes the incentives of the policy maker toexperiment. Such distrust mutes the “opportunistic” experimentation motive thatCogley, Colacito, and Sargent (2007) found to prevail especially when |U | is high.

4.3 Role of λ in determining worst-case submodel

Worst-case probabilities are context specific because they depend on the deci-sion maker’s objective function. To bring this out, we now explore how the pre-ceding results change as we increase the decision maker’s weight on inflation λ. Ahigher λ reduces the relative weight on unemployment in the period-loss functionand increases the weight on inflation. Therefore, it also alters the policy maker’sperceptions about worst-case scenarios.

When λ is 16, the policy maker cares more about inflation than unemployment,and the Samuelson-Solow model becomes the worst-case scenario. Figure 4, whichportrays the value function for the non-robust version of this model, shows that thevalue function now slopes downward along the α-axis, indicating that the authoritiesare better off when the Lucas model is true. When α = 0, they refrain from varyinginflation to stabilize unemployment, for they are unable to affect unemployment inany case, and they focus exclusively on maintaining price stability. That reducesinflation volatility at no cost in terms of higher unemployment volatility. Centralbankers who care mostly about inflation are happier in a classical world becausetheir job is easier in that environment.

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0.01

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−0.042

−0.04

−0.038

−0.036

−0.034

−0.032

−0.03

−0.028

−0.026

Prior on Samuelson and Solow

Unemployement

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Figure 4: Value function V (U, α) without robustness for λ = 16.

Figure 5 illustrates how this affects robust policies. If we were to hold θ2 constantwhile increasing λ, a concern for robustness would vanish for λ = 16, so we alsoreduce θ2 to 0.001 to compensate.11 Because the Samuelson-Solow model is the worstcase, a robust planner twists by increasing its probability weight. This explains whyin the left panel the twisted model weight α is greater than α in almost all statesof the world. This raises programmed inflation when unemployment is low, butbecause λ is so high, vt always remains close to zero, with or without robustness.Thus, differences in the policy functions are slight, amounting to just a few basispoints.

11When θ2 = 0.001 and λ = 0.1, the robust planner becomes hypervigilant, and the value functionceases to be concave in vt and convex in the choice of the perturbation to the approximating model.Whittle (1990) describes a breakdown value of θ1 as a point of ‘utter psychotic despair’.

13

α ≈ 0

0

5

10

x 10−4α = 0.2

Infla

tion

α = 0.4

0

5

10

x 10−4α = 0.6

Infla

tion

−.015 0 .015 .03

α = 0.8

Unemployment−.015 0 .015 .03

0

5

10

x 10−4α ≈ 1

Unemployment

Infla

tion

0

0.5

1

−0.01

0

0.01

0.02

0.03

0.95

1

1.05

1.1

1.15

1.2

1.25

αt

Ut

d(U t,α

t)

Figure 5: Robust policy with T2 operator only, with λ = 16 and θ2 = 0.001. In

the right panel, the dark line is the decision rule for the θ2 = +∞ no-robustnessdecision rule, while the grey line is the θ2 = 0.001 robust decision rule.

14

For intermediate values of λ, either model could be the worst, so the distortionto α could go either way. It follows that a concern for robustness could make policymore or less countercyclical. For example, figures 6 and 7 display the value function,α-distortion, and decision rules, respectively, for λ = 1 and θ2 = 0.001. Wheninflation and unemployment are equally weighted, the non-robust value functionstill slopes upward, which means that the Lucas model is still associated with theworst-case scenario, and the robust planner twists in most states of the world byreducing α relative to the Bayesian update α.12 Accordingly, the robust policy ruleis still less countercyclical than the Bayesian decision rule.

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−0.029

−0.028

−0.027

−0.026

−0.025

−0.024

−0.023

−0.022

Prior on Samuelson and SolowUnemployement

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Figure 6: Value function V (U, α) without robustness for λ = 1.

12An exception occurs when α is close to zero, where the robust planner twists toward theSamuelson-Solow model. This matters only slightly for policy because programmed inflation isalways close to zero when α is close to zero.

15

α ≈ 0

0

5

10

15x 10

−3α = 0.2

Infla

tion

α = 0.4

0

5

10

15x 10

−3α = 0.6

Infla

tion

−.015 0 .015 .03

α = 0.8

Unemployment−.015 0 .015 .03

0

5

10

15x 10

−3α ≈ 1

Unemployment

Infla

tion

0

0.5

1

−0.01

0

0.01

0.02

0.03

0.8

0.85

0.9

0.95

1

1.05

1.1

αt

Ut

d(U t,α

t)

Figure 7: Robust policy with T2 operator only, with λ = 1 and θ2 = 0.001. In the

right panel, the grey line is the decision rule for the θ2 = +∞ no-robustness decisionrule, while the dark line is the θ2 = 0.001 robust decision rule.

4.4 Dwindling effect of T2 operator

Colacito, Cogley, and Sargent (2007) indicate how, when one of the two submod-els is true, αt converges either to zero or one as t → +∞. Furthermore, even whenthe data are generated by a third submodel not considered by the decision maker, itis often the case that αt still converges to zero or one. The preceding figures indicatethat at the dogmatic boundary α = 0 or α = 1, there is no room for the T

2 operatorto distort beliefs. This means that the inexorable working of Bayes’ law causes theeffects of the T

2 operator to die off over time.

16

4.5 Activating T1: robustness with respect to each sub-

model

Keeping λ = 0.1, we now use the T1 operator to express a concern about misspec-

ification of the unemployment-inflation dynamics within each of the two submodels.To begin, by setting θ2 = +∞ we shall assume that the decision maker is confi-dent about his prior. We express a concern for misspecification of the submodels byreplacing the Es∗α∗ in (14) with the T

1 operator. In particular, we replace

β

∫V (Azst +Bzvt + Czǫt+1, πα(αt, Azst +Bzvt + Czǫt+1))dF (ǫt+1)

in (14) with (T1(βV )(st, αt, vt, z; θ1).Figures 8 and 9 display the conditional means and variances of the worst-case

conditional densities (22) for the Samuelson-Solow and Lucas models, respectively,for θ1 = 0.1 and θ2 = +∞13 The nature of the worst-case scenario is similar inthe two models. In both cases, the worst-case model envisions a higher probabilityof drawing a deviation-amplifying shock when |U | is already large. The expectedvalues of the distorted unemployment shocks in the two models, η1 and η2, arepositive when unemployment is high and negative when it is low, and this directlyamplifies unemployment volatility. Similarly, the expected values of the distortedinflation shocks, η3 and η4, are negative when U is high and positive when U islow. This indirectly increases unemployment volatility because U varies inverselywith respect to unexpected inflation. In addition, the shock variances are altered toincrease volatility, being greater when |U | is large.

Figure 10 displays the corresponding robust decision rule. To offset the greaterrisk of a deviation-amplifying shock, the robust policy authority adopts a more ag-gressive countercyclical stance relative to that set by a policy maker who fully truststhe specification of each model. Thus, concerns about possible misspecifications ofthe submodels have an opposite effect from a concern about the prior alone that wesummarized in figure 3.

13We set θ1 = .1 because this value delivers noticeable affects on the decision rule.

17

00.5

1

−0.010

0.010.02

0.03−0.04

−0.02

0

0.02

0.04

α

Distorted Et [η

1,t+1]

U 00.5

1

−0.010

0.010.02

0.03

1

1.01

1.02

α

Distorted Vart [η

1,t+1]

U

00.5

1

−0.010

0.010.02

0.03

−0.01

0

0.01

α

Distorted Et [η

3,t+1]

U 00.5

1

−0.010

0.010.02

0.03

1

1.001

1.002

α

Distorted Vart [η

3,t+1]

U

Figure 8: Conditional means and variances of distorted shocks to the Samuelson-Solow model with θ1 = .1.

18

00.5

1

−0.010

0.010.02

0.03−0.04

−0.02

0

0.02

0.04

0.06

α

Distorted Et [η

2,t+1]

U 00.5

1

−0.010

0.010.02

0.03

0.98

1

1.02

α

Distorted Vart [η

2,t+1]

U

00.5

1

−0.010

0.010.02

0.03

−0.01

0

0.01

α

Distorted Et [η

4,t+1]

U 00.5

1

−0.010

0.010.02

0.03

0.999

1

1.001

α

Distorted Vart [η

4,t+1]

U

Figure 9: Conditional means and variances of distorted shocks to the Lucas modelwith θ1 = .1.

19

α ≈ 0

−0.02

0

0.02

0.04

α = 0.2

Infla

tion

α = 0.4

−0.02

0

0.02

0.04

α = 0.6

Infla

tion

−0.01 0 0.01 0.02 0.03

α = 0.8

Unemployment−0.01 0 0.01 0.02 0.03

−0.02

0

0.02

0.04

α ≈ 1

Unemployment

Infla

tion

Figure 10: Robust policy with T1 operator only, θ1 = .1. The grey line is the decision

rule for the θ1 = +∞ no-robustness decision rule, while the dark line is the θ1 = 0.1robust decision rule.

20

4.6 How the two forms of misspecification interact: activat-

ing both the T1 and T

2 operators

Figure 11 activates concerns about both features of the specification. As mightbe guessed from the complexion of the earlier results, turning on both sources ofconcern about robustness yields a decision rule that is close to the one we obtainedwithout any concerns about robustness. When λ = 0.1, the decision maker makesprogrammed inflation less countercyclical to guard against misspecification of theprior, but makes vt more countercyclical to protect against misspecification of thetwo submodels. In effect, the worst-case α shown in the left panel of figure 11 offsetsthe worst-case dynamics coming from the dependence of the worst-case conditionalmean on (Ut, αt), so that the combined effects of T

1 and T2 approximately cancel.

Thus, the optimal Bayesian decision rule with experimentation – calculated withoutexplicit reference to robustness – is robust to a mixture of concerns about the twotypes of misspecification.14

α ≈ 0

0

0.02

0.04

α = 0.2

Infla

tion

α = 0.4

0

0.02

0.04

α = 0.6

Infla

tion

−.015 0 .015 .03

α = 0.8

Unemployment−.015 0 .015 .03

0

0.02

0.04

α ≈ 1

Unemployment

Infla

tion

0

0.5

1

−0.01

0

0.01

0.02

0.03

0.95

0.96

0.97

0.98

0.99

1

αt

Ut

d(U t,α

t)

Figure 11: Worst-case α and decision rule with concerns about both source of mis-specification, captured by T

1 and T2 with θ1 = θ2 = .1. The black line on the

right panel indicates the θ1 = θ2 = +∞ decision rule and the grey line indicates theθ1 = θ2 = .1 decision rule.

14Results like this also obtain for other values of λ.

21

4.7 How the two operators influence experimentation

Next we examine more closely how the two risk-sensitivity operators affect mo-tives to experiment. Figure 12 compares robust, Bayesian, and anticipated-utilitydecision rules. To highlight their differences, we set α = 0.4 to focus on a partof the state space where experimental motives are strongest. We interpret differ-ences of robust decision rules relative to the non-experimental, anticipated-utilitydecision rule. When a risk-sensitivity operator moves a decision rule closer to theanticipated-utility policy, we say that it tempers experimentation.

−0.01 0 0.01 0.02 0.03−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06α = 0.4

−0.01 0 0.01 0.02 0.03−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06α = 0.4

−0.01 0 0.01 0.02 0.03−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06α = 0.4

Anticipated Utilityθ

1=∞, θ

2=0.1

θ1=∞, θ

2=∞

Anticipated Utilityθ

1=0.1, θ

2=∞

θ1=∞, θ

2=∞

Anticipated Utilityθ

1=0.1, θ

2=0.1

θ1=∞, θ

2=∞

Figure 12: Robust, Bayesian, and Anticipated-Utility Policy Rules

The left panel of figure 12 illustrates the influence of the backward-looking T2

operator. On balance, T2 mutes experimentation. For small values of |U |, the

robust and Bayesian policies are essentially the same, while for larger values therobust policy curls back toward the nonexperimental decision rule. Since the robustrule calls for no more experimentation than the Bayesian policy when unemploymentis close to the natural rate and calls for less when the unemployment gap is large inmagnitude, less experimentation occurs along a learning transition path.15

The middle panel examines the influence of the forward-looking T1 operator. In

this case, experimentation is muted for small values of |U | but strongly enhanced for

15It is conceivable that T2 results in more experimentation for larger values of |U | not shown on

the graph, but those states are rarely visited.

22

large values. Since realizations of |U | in the neighborhood of 0.02 are not unusual,T1

typically results in more experimentation.Finally, the right-hand panel illustrates what happens when both operators are

active. Since one operator mutes and the other enhances experimentation, the twooperators offset, so that a decision rule coming from the application of both operatorsis about the same as the Bayesian decision rule. In summary, the net effect ofrobustness on experimentation is ambiguous and depends on the penalty parametersθ1 and θ2.

4.8 How long does it take to learn the truth?

In this subsection we analyze how a preference for robustness affects the numberof quarters that are needed to learn the true model. We address this question bysimulation. For each simulation, we assume that either the Lucas’ model or theSamuelson and Solow model is the actual data generating process. We initializethe state space at various levels of (α0, U0) and let the system run according to thedynamics of the true model under the optimal inflation rate impelled by the relevantBellman equation. For each experiment we report the median number of quarters,that are needed for α to get within a 0.01 neighborhood of what it should be underthe true data generating process. We also report the 10%-90% confidence intervalfor each case. Each experiment is based on 1000 simulations of length 700 quarters.The results are reported in table 1, and they can be summarized by comparison tothe baseline case (i.e. (λ = 0.1,θ1 = +∞, θ2 = +∞)).

1. A fear for prior misspecification (i.e., θ1 = +∞, θ2 = 0.1) increases the timeneeded to learn the true model. This is particularly apparent for high initiallevels of unemployment. In these cases, the distorted probability distributionmakes the Samuelson and Solow model less likely. Hence, the optimal decisionrule calls for a lower inflation rate, which damps experimentation and makesit harder to discover the actual data generating process (see figure 3).

2. A fear of misspecification of the probability distribution within each sub-model(i.e., θ1 = 0.1, θ2 = +∞) increases the speed of convergence. In this case, thepolicy maker sets higher rates of inflation in the rise of unemployment (seefigure 8). The higher the degree of experimentation results in a usually veryquick convergence to the true model.

3. When both risk-sensitivity operators are turned on, there is no significantdifference with respect to the baseline model. This is the result of the two fearsof misspecification offsetting each other in the choice of the optimal inflationrate (see figure 11).

23

Waiting time

True Model α0 U0θ1 = +∞ θ1 = +∞ θ1 = 0.1 θ1 = 0.1θ2 = +∞ θ2 = 0.1 θ2 = +∞ θ2 = 0.1

SS 0.01 0 218 223 214 231[128,439] [130,444] [123,416] [128,451]

SS 0.01 0.025 229 233 224 241[141,459] [135,453] [143,404] [147,441]

Lucas 0.99 0 89 96 66 89[32,197] [39,204] [22,150] [34,202]

Lucas 0.99 0.025 66 80 54 71[20,175] [27,203] [4,149] [19,194]

SS 0.5 0 40 37 38 37[21,79] [20,75] [18,71] [20,70]

SS 0.5 0.025 22 27 9 20[5,61] [10,73] [2,51] [4,53]

Lucas 0.5 0 72 78 60 74[26,192] [26, 180] [17,160] [23,188]

Lucas 0.5 0.025 58 71 28 59[15,167] [23,187] [2,127] [14,177]

Table 1: Waiting Times (in quarters) for various data-generating processes andinitial (α0, U0) pairs. The variable that we call waiting time represents the numberof quarters that are needed for α to return to within a 0.01 neighborhood of whatit should be under the data generating process. For each experiment, we report thetrue model, the initial prior, the initial unemployment rate, the median waiting time,and the 10% - 90% confidence sets in square brackets for various pairs of (θ1, θ2).

24

5 Concluding remarks

In this paper, we study how concerns for robustness modify incentives to ex-periment. We use a decision theory that explores robustness of decision rules bycalculating bounds on value functions over a set of probability models near a deci-sion maker’s approximating model. Seeking bounds on value functions over a set ofprobability models automatically leads to a worst-case analysis. We study a settingin which a decision maker’s approximating model is an average of two submodels.The decision maker uses Bayes’ law to update priors over submodels as new data ar-rive. Our T

1 operator checks robustness of each submodel. Our T2 operator checks

robustness with respect to a prior over submodels.Our working example is the model in Colacito, Cogley, and Sargent (2007) in

which a Samuelson-Solow submodel offers a permanently exploitable trade-off be-tween inflation and unemployment and another Lucas submodel lacks a trade-offthat is even temporarily exploitable. This is a good setting for illustrating howthe worst-case model is worst relative to the decision maker’s objective. When themonetary policy decision maker puts more weight on unemployment (λ = 0.1), theLucas model is worse. That makes the robust policy less countercyclical than thepolicy that completely trusts the model. When more weight is on inflation (λ = 16),the Samuelson-Solow model is worse for the policy maker. That makes the robustpolicy more countercyclical than the nonrobust policy.

Robust policy makers have an incentive to experiment for the same reason thatBayesian policy makers do. The decision maker’s posterior is still an element of thestate vector, so robust Bellman equations continue to instruct the decision maker toexperiment with an eye toward tightening the posterior in the future. What changesare the costs and benefits of experimentation. How decision rules are altered is modelspecific, so robustness could in principle either enhance or mute experimentation.

In the present context, the T1 and T

2 operators have countervailing effects onpolicy. When λ = 0.1, concerns that the submodels are misspecified make policymore countercyclical than in a Bayesian setting, while concerns that the prior ismisspecified make policy less countercyclical. When these results are compared toCogley, Colacito, and Sargent’s (2007) measures of the contribution of an exper-imentation motive to the policy rule, they show that with complete trust in theprior over submodels, distrust of the submodels increases the motive to experiment,while with complete trust in the submodels, distrust of the prior over submodelsdiminishes the motive to experiment. When both operators are active, their effectsapproximately cancel, and the robust policy well approximates the Bayesian decisionrule. Since Cogley, Colacito, and Sargent (2007) showed that the Bayesian decisionrule well approximates an ‘anticipated-utility’ policy that suppresses experimenta-tion altogether, it follows that with both of our T operators active, the optimally

25

robust policy has little or no experimentation. Thus, the disagreement between theordinary Bellman equation’s recommendation to experiment and Blinder’s and Lu-cas’s advice not to experiment, cited at the beginning of this paper, can in principlebe rationalized by using the T

2 operator to express a distrust of the decision maker’sprior over the submodels that offsets other motives to experiment.

A Details

A.1 The Bellman equation

Our Bellman equation without fear of model misspecification is:

V (st, αt) = maxvt

r(st, vt) (14)

+ βαt

∫V (A1st +B1vt + C1ǫt+1, πα(αt, A1st +B1vt + C1ǫt+1))dF (ǫt+1)

+ β(1 − αt)

∫V (A2st +B2vt + C2ǫt+1, πα(αt, A2st +B2vt + C2ǫt+1))dF (ǫt+1)

The optimal decision rule can be represented recursively as

vt = v(st, αt). (15)

Repeated substitution of (7) into (15) yields the policy maker’s strategy in the formof a sequence of functions

vt = σt(st, α0). (16)

Cogley, Colacito, and Sargent (2007) derive the function πα(st, αt). To summarizetheir calculations, let Ωi = CiC

′i, Rt = αt

1−αt

, and define

g(ǫt+1; st, αt) = logRt −1

2log |Ω1| +

1

2log |Ω2| −

1

2(C1ǫt+1)

′ Ω−11 (C1ǫt+1)

+1

2[(A1 −A2)st + (B1 − B2)vt + C1ǫt+1]

′

× Ω−12 [(A1 −A2)st + (B1 − B2)vt + C1ǫt+1] (17)

and

h(ǫt+1; st, αt) = logRt −1

2log |Ω1| +

1

2log |Ω2| +

1

2(C2ǫt+1)

′ Ω−12 (C2ǫt+1)

−1

2[(A2 − A1)st + (B2 − B1)vt + C2ǫt+1]

′

× Ω−11 [(A2 − A1)st + (B2 − B1)vt + C2ǫt+1]. (18)

26

The Bellman equation (14) becomes

V (st, αt) = maxvt

r(st, vt) + βαt

∫V

(A1st + B1vt + C1ǫt+1,

eg(ǫt+1)

1 + eg(ǫt+1)

)dF (ǫt+1)

+β(1 − αt)

∫V

(A2st + B2vt + C2ǫt+1,

eh(ǫt+1)

1 + eh(ǫt+1)

)dF (ǫt+1)

. (19)

Cogley, Colacito, and Sargent (2007) also describe how to approximate the solutionof (19) and the robust counterpart to it that we propose in subsection 3.2.

A.2 The two operators

We describe details of how the operators T1,T2 apply in our particular setting.

A.3 T1 operator

For a given value function V (st+1, αt+1) and a given decision rule vt = v(st, αt),define

T1(V (st+1, αt+1)((st, αt, vt, z; θ1) (20)

= −θ1 log

∫exp

(−V (Azst +Bzvt + Czǫt+1, πα(αt, Azst +Bzvt + Czǫt+1))

θ1

)dF (ǫt+1).

= minφ(st,vt,αt,ǫt+1)≥0

∫ [V (Azst +Bzvt + Czǫt+1, πα(αt, Azst +Bzvt + Czǫt+1))

+ θ1 log φ(st, vt, αt, ǫt+1)]φ(st, vt, αt, ǫt+1)dF (ǫt+1) (21)

where the minimization is subject to E[φ(st, vt, αt, ǫt+1)|st, αt, vt, j] = 1. The mini-mizer in (21) is a worst-case distortion to the density of ǫt+1:

φ∗(ǫt+1, st, αt) =exp(−V (Azst+Bzvt+Czǫt+1,πα(αt,Azst+Bzvt+Czǫt+1))

θ1

)

∫exp(−V (Azst+Bzvt+Czǫt+1,πα(αt,Azst+Bzvt+Czǫt+1))

θ1

)dF (ǫt+1)

,

where it is understood that vt on the right side is evaluated at a particular decisionrule v(st, αt). The distorted conditional density of ǫt+1 is then

φ(ǫt+1, st, αt) = φn(ǫt+1)φ∗(ǫt+1, st, αt), (22)

where φn(ǫt+1) is the standard normal density.

27

A.4 T2 operator

For j = 1, 2, let V (s, α, v, z) be distinct functions of (s, α, v) for z = 0, 1. Define

T2(V (s, α, v, z; θ2)(s, α, v) (23)

= −θ2 log[α exp

(−V (s, α, v, 0)

θ2

)+ (1 − α) exp

(−V (s, v, α, 1)

θ2

)]

= minψ0≥0,ψ1≥0

[V (s, α, v, 0) + θ2 logψ1]ψ0α

+ [V (s, α, v, 1) + θ2 logψ1]ψ1(1 − α)

(24)

where the minimization is subject to ψ0α + ψ1(1 − α) = 1. The minimizers of (24)are

ψ∗0(s, α, v) = k exp

(−V (s, v, α, 0)

θ2

)

ψ∗1(s, α, v) = k exp

(−V (s, v, α, 1)

θ2

)

where k−1 = exp(

−V (s,α,v,0)θ2

)α + exp

(−V (s,v,α,1)

θ2

)(1 − α). An associated worst-case

probability that z = 0 is given by

α = ψ∗0(s, α, v)α. (25)

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29